Extremal Homogeneous Polynomials on Real Normed Spaces

If Pis a continuous m-homogeneous polynomial on a real normed space and Pis the associated symmetric m-linear form, the ratio ‖ P‖/‖ P‖ always lies between 1 and m m / m!. We show that, as in the complex case investigated by Sarantopoulos (1987, Proc. Amer. Math. Soc. 99, 340–346), there are P's for which ‖ P‖/‖ P‖= m m / m! and for which Pachieves norm if and only if the normed space contains an isometric copy of ℓ m 1. However, unlike the complex case, we find a plentiful supply of such polynomials provided m⩾4.